multiply vector by vector

Daniel Parker

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17-03-2025

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Multiplying vectors isn't as straightforward as multiplying numbers. There are actually two main ways to "multiply" vectors, each with its own meaning and application: the dot product (scalar product) and the cross product (vector product). This article will explain both, providing examples and clarifying their differences.

Understanding Vector Multiplication: Dot Product

The dot product, also known as the scalar product, results in a single number (a scalar), not another vector. This number represents the projection of one vector onto another, scaled by the magnitude of the other vector. It's particularly useful in calculating work, angles between vectors, and other scalar quantities derived from vector interactions.

Calculating the Dot Product

The dot product of two vectors a and b is calculated as follows:

a • b = |a| |b| cos θ

where:

• |a| and |b| represent the magnitudes (lengths) of vectors a and b, respectively.
• θ is the angle between the two vectors.

Alternatively, if the vectors are represented in component form (e.g., a = <a₁, a₂> and b = <b₁, b₂> in two dimensions, or a = <a₁, a₂, a₃> and b = <b₁, b₂, b₃> in three dimensions), the dot product can be computed as the sum of the products of corresponding components:

a • b = a₁b₁ + a₂b₂ (in two dimensions)

a • b = a₁b₁ + a₂b₂ + a₃b₃ (in three dimensions)

Example: Dot Product Calculation

Let's say we have two vectors: a = <2, 3> and b = <4, 1>. The dot product is:

a • b = (2 * 4) + (3 * 1) = 8 + 3 = 11

Applications of the Dot Product

• Calculating Work: In physics, the work done by a force F on an object moving through a displacement d is given by the dot product: W = F • d.
• Determining Orthogonality: If the dot product of two vectors is zero, the vectors are orthogonal (perpendicular) to each other.
• Finding the Angle Between Vectors: The formula for the dot product can be rearranged to find the angle θ between two vectors: cos θ = (a • b) / (|a| |b|)

Understanding Vector Multiplication: Cross Product

Unlike the dot product, the cross product (also called the vector product) results in a new vector. This new vector is perpendicular to both of the original vectors. Its direction is determined by the right-hand rule, and its magnitude represents the area of the parallelogram formed by the two original vectors. The cross product is primarily used in physics and engineering, particularly in areas involving torque, angular momentum, and magnetic fields.

Calculating the Cross Product

The cross product of two vectors a and b is denoted as a x b. In three dimensions, it's calculated using the determinant of a 3x3 matrix:

a x b = | | | <a₁, a₂, a₃> | | <b₁, b₂, b₃> |